Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #37761 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(25,26,32,36)(27,37,39,33)(28,30,29,34)(31,35,40,38);(3,7)(6,8)(9,10,14,15)(11,13,17,22)(12,19,21,16)(18,24,23,20)(25,27,32,39)(26,33,36,37)(28,38,29,35)(30,40,34,31);(1,2,5,4)(3,6,7,8)(10,15)(13,22)(16,19)(20,24)(25,28,32,29)(26,34,36,30)(27,38,39,35)(31,37,40,33);(1,3,5,7)(2,6,4,8)(9,11,14,17)(10,13,15,22)(12,18,21,23)(16,20,19,24)(25,29,32,28)(26,30,36,34)(27,38,39,35)(31,37,40,33);(1,4,5,2)(3,8,7,6)(9,12,14,21)(10,16,15,19)(11,18,17,23)(13,20,22,24)(25,30)(26,28)(27,31)(29,36)(32,34)(33,35)(37,38)(39,40);(9,13)(10,17)(11,15)(12,20)(14,22)(16,23)(18,19)(21,24)(25,31)(26,35)(27,30)(28,33)(29,37)(32,40)(34,39)(36,38)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,d^2=hi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8135620);